Small Lattice Spacing Research Articles

Characteristic of improved fatigue performance for Zr-based bulk metallic glass matrix composites

Zr58.5Ti14.3Nb5.2Cu6.1Ni4.9Be11.0 bulk metallic glass matrix composites exhibit improved four-point-bending fatigue endurance with a fatigue limit of 567MPa, compared to that under the tension–tension fatigue, due to the high-volume-fractioned dendrites, which can effectively blunt the fatigue-induced cracks. Illuminated by high-energy synchrotron X-ray at 200 and 100K, the corresponding diffraction peaks, such as (110), (200), and (211) shift rightward to small lattice spacings, compared to those at 298K. However, the peak widths at 100K and 200K are almost identical to that of room temperature. Since an identical fatigue specimen was measured under room temperature, 200K, and 100K, the invariant of the peak widths reveal the fact of the irreversible microstructure developments induced by fatigue. Even if the fatigue fracture stress is distinguishingly lower than the yielding strength, the deformation of dendrites locally prevails, evidenced by the occurrence of dislocations.

Lattice QCD with open boundary conditions and twisted-mass reweighting

Lattice QCD simulations at small lattice spacings and quark masses close to their physical values are technically challenging. In particular, the simulations can get trapped in the topological charge sectors of field space or may run into instabilities triggered by accidental near-zero modes of the lattice Dirac operator. As already noted in ref. [1], the first problem is bypassed if open boundary conditions are imposed in the time direction, while the second can potentially be overcome through twisted-mass determinant reweighting [2]. In this paper, we show that twisted-mass reweighting works out as expected in QCD with open boundary conditions and 2+1 flavours of O(a) improved Wilson quarks. Further algorithmic improvements are tested as well and a few physical quantities are computed for illustration.

Individual eigenvalue distributions for the Wilson Dirac operator

We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D5 as well as for real eigenvalues of the Wilson Dirac Operator DW. The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours Nf and for non-zero low energy constants W6, W7, W8. It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of DW at fixed chirality nu this expansion truncates after at most nu terms for small lattice spacing "a". Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D5, where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched DW at small "a" we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size nu.

Are there hadronic bound states above the QCD transition temperature?

Recent lattice QCD calculations, at physical pion masses and small lattice spacings that approach the continuum limit, have revealed that non-diagonal quark correlators above the critical temperature are finite up to about 2 $T_c$. Since the transition from hadronic to free partonic degrees of freedom is merely an analytic cross-over, it is likely that, in the temperature regime between 1-2 $T_c$, quark and gluon quasiparticles and pre-hadronic bound states can coexist. The correlator values, in comparison to PNJL model calculations beyond mean-field, indicate that at least part of the mixed phase resides in color-neutral bound states. A similar effect was postulated for the in-medium fragmentation process, i.e. for partons which do not thermalize with the system and thus constitute the non-equilibrium component of the particle emission spectrum from a deconfined plasma phase. Here, for the first time we investigate the likelihood of forming bound states also in the equilibrated, parton dominated phase above $T_c$ which is described by lattice QCD.

Closed flux tubes and their string description in D=2+1 SU(N) gauge theories

We carry out lattice calculations of the spectrum of confining flux tubes that wind around a spatial torus of variable length l, in 2+1 dimensions. We compare the energies of the lowest c.30 states to the free string Nambu-Goto model and to recent results on the universal properties of effective string actions. Our most useful calculations are in SU(6) at a small lattice spacing, which we check is very close to the large-N continuum limit. We find that the energies, En(l), are remarkably close to the predictions of the free string Nambu-Goto model, even well below the critical length at which the expansion of the Nambu-Goto energy in powers of 1/l diverges and the series needs to be resummed. Our analysis of the ground state supports the universality of the O(1/l) and the O(1/l^3) corrections to l.sigma, and we find that the deviations from Nambu-Goto at small l prefer a leading correction that is O(1/l^7), consistent with theoretical expectations. We find that the low-lying states that contain a single phonon excitation are also consistent with the leading O(1/l^7) correction dominating down to the smallest values of l. By contrast our analysis of the other light excited states clearly shows that for these states the corrections at smaller l resum to a much smaller effective power. Finally, and in contrast to our recent calculations in D=3+1, we find no evidence for the presence of any non-stringy states that could indicate the excitation of massive flux tube modes.

Probing the chiral regime ofNf=2QCD with mixed actions

We report on our first experiences with a mixed action setup with overlap valence quarks and non-perturbatively O(a) improved Wilson sea quarks. For the latter we employ CLS Nf=2 configurations with light sea quark masses at small lattice spacings. Exact chiral symmetry allows to consider very light valence quarks and explore the matching to (partially quenched) Chiral Perturbation Theory (ChPT) in a mixed epsilon/p-regime. We compute the topological susceptibility and the low-lying spectrum of the massless Neuberger-Dirac operator for three values of the sea quark mass, and compare the sea quark mass dependence to NLO ChPT in the mixed regime. This provides two different determinations of the chiral condensate, as well as information about some NLO low-energy couplings. Our results allow to test the consistency of the mixed-regime approach to ChPT, as well as of the mixed action framework.

The multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test: I

We propose an algorithmic procedure (i) to study the ‘distance’ between an integrable PDE and any discretization of it, in the small lattice spacing ϵ regime, and, at the same time, (ii) to test the (asymptotic) integrability properties of such discretization. This method should provide, in particular, useful and concrete information on how good is any numerical scheme used to integrate a given integrable PDE. The procedure, illustrated on a fairly general ten-parameter family of discretizations of the nonlinear Schrödinger equation, consists of the following three steps: (i) the construction of the continuous multiscale expansion of a generic solution of the discrete system at all orders in ϵ, following Degasperis et al (1997 Physica D 100187–211); (ii) the application, to such an expansion, of the Degasperis–Procesi (DP) integrability test (Degasperis A and Procesi M 1999 Asymptotic integrability Symmetry and Perturbation Theory, SPT98, ed A Degasperis and G Gaeta (Singapore: World Scientific) pp 23–37; Degasperis A 2001 Multiscale expansion and integrability of dispersive wave equations Lectures given at the Euro Summer School: ‘What is integrability?’ (Isaac Newton Institute, Cambridge, UK, 13–24 August); Integrability (Lecture Notes in Physics vol 767) ed A Mikhailov (Berlin: Springer)), to test the asymptotic integrability properties of the discrete system and its ‘distance’ from its continuous limit; (iii) the use of the main output of the DP test to construct infinitely many approximate symmetries and constants of motion of the discrete system, through novel and simple formulas.

Chiral properties of SU(3) sextet fermions

SU(3) gauge theory with overlap fermions in the 2-index symmetric (sextet) and fundamental representations is considered. A priori it is not known what the pattern of chiral symmetry breaking is in a higher dimensional representation although the general expectation is that if two representations are both complex, the breaking pattern will be the same. This expectation is verified for the sextet at N_f = 0 in several exact zero mode sectors. It is shown that if the volume is large enough the same random matrix ensemble describes both the sextet and fundamental Dirac eigenvalues. The number of zero modes for the sextet increases approximately 5-fold relative to the fundamental in accordance with the index theorem for small lattice spacing but zero modes which do not correspond to integer topological charge do exist at larger lattice spacings. The zero mode number dependence of the random matrix model predictions correctly match the simulations in each sector and each representation.

QCD transition temperature: approaching the continuum on the lattice

In order to clarify the source of the discrepancy between our previous transition temperature determination [Phys. Lett. B 643, 46 (2006)] and that of the Bielefeld-Brookhaven-Columbia-Riken collaboration we improved our calculations by taking even smaller lattice spacings ( N t = 12 and N t = 16 at one point) and by using physical quark masses also for the T = 0 quantities. In addition to the kaon decay constant used for scale setting we determine four quantities (masses of the Ω baryon, k*(892) and ϕ(1020) mesons and the pion decay constant) which are found to agree with experiment. This implies that – independently of which of these quantities is used to set the overall scale – the same results are obtained within a few percent. At finite temperature we use finer lattices down to a ≲ 0.1 fm . Our new results confirm completely our previous findings. We compare the results with those of the ‘hotQCD’ collaboration.

Spatial string tension in N f = 2 lattice QCD at finite temperature

The spatial string tension across a crossover from the low temperature phase to the high temperature phase is computed in QCD with two flavors of non-perturbatively improved Wilson fermions at small lattice spacing a \sim 0.12fm. We find that in the low temperature phase spatial string tension agrees well with zero temperature string tension. Furthermore, it does not show increasing for temperatures up to T = 1.36 T_{pc}, the highest temperature considered. Our results agree with some theoretical predictions.

Van der Waals energy of a 1-dimensional lattice

Van der Waals dispersion forces are examined for an infinite one-dimensional (1D) chain of atoms. Building on the work of Bade and Kirkwood (J. Chem. Phys. 27, 1284, 1957), dipolar fluctuations of the 1D chain are analyzed in terms of polarization waves. The van der Waals energy of the chain is evaluated from the zero-point energy of these waves. Several results are found. (1) A critical value exists for the dimensionless coupling parameter ν = α0/a 3, for large polarizabilities (α0) or small lattice spacings (a). When ν > 0.208, the 1D line has divergences in both the dispersion relation of the waves and the effective site polarizability. The resulting instability is analogous to 1D ferroelectricity. (2) The sum of higher order (than three-body) interactions are shown to be more important than three-body interactions (which are small); when ν approaches 0.208, the higher order terms are comparable to the two-body term. (3) Finite 1D chains of N atoms behave like infinite chains when N exceeds 100. (4) 1D surface tensions deviate from two-body predictions for large values of the coupling parameter (ν > 0.10).

Modification of directed-loop algorithm for continuous space simulation of bosonic systems

We modify the directed-loop algorithm (DLA) to solve the problem that typically arises from large on-site interaction. The large on-site interaction is inevitable when one tries to simulate a Bose gas system in continuum by discretizing the space with small lattice spacings. While the efficiency of a straightforward application of DLA decreases as the mesh becomes finer, the performance of the new method does not depend on it except for the trivial factor due to the increase in the number of lattice points.

A new lattice measurement for potentials between static SU(3) sources

In this article, a new calculation of static potentials between sources of different representations in SU(3) gauge group is presented. The results of author's previous study \cite{Deld00} at the smallest lattice spacing $a_{s}\simeq0.11$~ fm are shown to have been affected by finite volume effects. Within statistical errors, the new results obtained here are still in agreement with both, Casimir scaling and flux tube counting. There is also no contradiction to the results obtained in Ref.~ \cite{Bali00} which however exclude flux counting.

Three dimensionalN= 2 supersymmetry on the lattice

We show how 3-dimensional, N=2 supersymmetric theories, including super QCD with matter fields, can be put on the lattice with existing techniques, in a way which will recover supersymmetry in the small lattice spacing limit. Residual supersymmetry breaking effects are suppressed in the small lattice spacing limit by at least one power of the lattice spacing a.

Topology of [formula omitted] gauge theories at [formula omitted] and [formula omitted

We calculate the topological charge density of SU ( N ) lattice gauge fields for values of N up to N = 8 . The calculations are performed mostly at a fixed lattice spacing, a ≃ 1 / 5 T c , at which most other physical quantities show small lattice spacing corrections. Our T ≃ 0 topological susceptibility appears to approach a finite non-zero limit at N = ∞ that is consistent with earlier extrapolations from smaller values of N. Near the deconfining temperature T c , we are able to investigate separately the confined and deconfined phases, since the transition is quite strongly first order. We find that the topological susceptibility in the confined phase is very similar at all T to that at T = 0 . By contrast, in the deconfined vacuum at larger N there are no topological fluctuations except for rare, isolated and small instantons. This shows that as N → ∞ the large- T suppression of large instantons and the large- N suppression of small instantons overlap, even at T ≃ T c , so as to suppress a l l topological fluctuations in the deconfined phase. In the confined phase by contrast, the size distribution is much the same at all T, becoming more peaked as N grows, suggesting that perhaps D ( ρ ) ∝ δ ( ρ − ρ c ) at N = ∞ , with ρ c ∼ 1 / T c .

QCD phase diagram: A comparison of lattice and hadron resonance gas model calculations

We compare the lattice results on QCD phase diagram for two and three flavors with the hadron resonance gas model (HRGM) calculations. Lines of constant energy density $ϵ$ have been determined at different baryo-chemical potentials ${\ensuremath{\mu}}_{B}$. For the strangeness chemical potentials ${\ensuremath{\mu}}_{S}$, we use two models. In one model, we explicitly set ${\ensuremath{\mu}}_{S}=0$ for all temperatures and baryo-chemical potentials. This assignment is used in lattice calculations. In the other model, ${\ensuremath{\mu}}_{S}$ is calculated in dependence on $T$ and ${\ensuremath{\mu}}_{B}$ according to the condition of vanishing strangeness. We also derive an analytical expression for the dependence of ${T}_{c}$ on ${\ensuremath{\mu}}_{B}/T$ by applying Taylor expansion of $ϵ$. In both cases, we compare HRGM results on ${T}_{c}\ensuremath{-}{\ensuremath{\mu}}_{B}$ diagram with the lattice calculations. The agreement is excellent, especially when the trigonometric function of $ϵ$ is truncated up to the same order as done in lattice simulations. For studying the efficiency of the truncated Taylor expansion, we calculate the radius of convergence. For zero- and second-order radii, the agreement with lattice is convincing. Furthermore, we make predictions for QCD phase diagram for nontruncated expressions and physical masses. These predictions are to be confirmed by heavy-ion experiments and future lattice calculations with very small lattice spacing and physical quark masses.

Properties of the large N phase transition

Reduced large N gauge theories have a transition between phases with unbroken and broken center symmetry. The phase with unbroken symmetry is associated with the zero temperature, infinite volume, infinite N theory. Measurements of the scalling properties of the latent heat test the hypothesis that the phase in which the symmetry is broken in just one spatial direction is the reduction of the high temperature, infinite volume, infinite N theory. While substantial finite spacing effects are observed, physical scaling for the latent heat at smaller lattice spacing is a viable possibility.

Impact of large cutoff-effects on algorithms for improved Wilson fermions

As a feasibility study for a scaling test we investigate the behavior of algorithms for dynamical fermions in the N f = 2 Schrödinger functional at an intermediate volume of 1 fm 4 . Simulations were performed using HMC with two pseudo-fermions and PHMC at lattice spacings of approximately 0.1 and 0.07 fm . We show that some algorithmic problems are due to large cutoff-effects in the spectrum of the improved Wilson–Dirac operator and disappear at the smaller lattice spacing. The problems discussed here are not expected to be specific to the Schrödinger functional.

Staggered eigenvalue mimicry

We study the infrared part of the spectrum for UV-filtered staggered Dirac operators and compare them to the overlap counterpart. With sufficient filtering and at small enough lattice spacing the staggered spectra manage to ``mimic'' the overlap version. They show a 4-fold near-degeneracy, and a clear separation between would-be zero modes and non-zero modes. This suggests an approximate index theorem for filtered staggered fermions and a correct sensitivity to the topology of QCD. Moreover, it supports square-rooting the staggered determinant to obtain dynamical ensembles with $N_f=2$.

Calculation of kaon matrix elements in quenched domain-wall QCD with DBW2 gauge action

We give a progress report of our new $a^{-1}\approx 3$ GeV quenched calculation of kaon matrix elements with domain-wall fermion and DBW2 gauge action. Our smaller lattice spacing allows us to address the effect of charmed quark on the lattice. We show preliminary results of $B_K$ renormalized non-perturbatively and $K\to\pi$ matrix elements.